1. Deductively Definable
Logics of Induction

The class of logics I will investigate are "deductively
definable." They are based on the inductive strengths

[A|B]

which we will read as "the strength of support that
proposition A accrues from proposition B." The strength [A|B] will often be a
real number, such as a probability, but it need not be.

An inductive logic is the set of rules used to assign these
strengths. In a deductively definable logic of
induction, these rules employ only deductive relations between
propositions.

[A|B]
=

some formula that
mentions only
deductive relations
between propositions

You might think that such a logic is exceptional. They are
not. Indeed one of the most common ways of specifying inductive relations in
the literature on inductive inference is through deductive relations.

The idea is a natural one. In its crudest form, inductive
inference is introduced as some variant of
deductive inference. One way of seeing induction is as a form of
partial entailment. If the truth of B necessitates the truth of
A, then we have full deductive entailment. If, however, it succeeds only
partially, then we have inductive support. Or, in another approach, inductive
inference is seen as an inversion of deductive inference. If A deductively
entails B, then B provides inductive support for A.

Elsewhere in "A Little Survey of
Induction," I have described how the many accounts of inductive
inference can be divided into three general
families. The first two are "inductive generalization" and
"hypothetical induction." Both depend upon characterizing inductive support
in terms of deductive relations.

Inductive generalization
depends upon the notion of an instance of a generalization. The most familiar
example arises in syllogistic logic. The proposition that "This A is B." is
an instance of the generalization "All A's are B." Hempel famously extended
this notion of an instance to first order predicate logic. All logics in the
inductive generalization family depend in one form or another on the rule

Instance I confirms generalization G Â Â if Â Â I
is an instance of G.

This rule lies within deductively definable logics since
the essential clause on the right of the definition "I is an instance of G"
specifies a deductive relation between propositions. This fact continues to
hold for the more elaborate versions of instance confirmation, such as
Glymour's "bootstrap."

Once again, the rule lies within deductively definable
logics since the essential clause on the right of the definition, "H
deductively entails E," specifies a deductive relation between
propositions.

This simplest form of hypothetical deduction is generally
found to be too simple. There are many proposals to amend it. Generally, the
amendments require that H not merely entail E,
but that it do it in the right way. What constitutes the "right way" is the
subject of continuing investigations. One proposal requires that it do so
simply. Another requires that H not just entail E but that it
explain E. Yet another requires that the entailment not be ad
hoc. All these amount to the addition of an extra condition "D" to form
the new rule:

This augmented rule will still lie within the deductively definable logics as long as the condition D
is itself a condition expressed fully in terms of the deductive relations
between the propositions. It is certainly plausible, but perhaps not
necessary, that notions like simplicity and explanatory character can be
specified by deductive relations. The simpler deduction, for example, might
merely employ fewer steps. Or the more explanatory hypothesis might infer the
evidence from some small set of propositions that have so many deductive
consequences that we single them out as "laws." Ad hocness is generally
characterized in terms of the order in which we infer things. If H entails an
already know E, H may not get as much support as it would if it were to
entail evidence not already known. This notion of ad hocness can be captured
if we expand our logic to include a time indexing of propositions that
indicates the order in which we learned them.

The rules mentioned so far enable
the assigning of a single value to [H|E]. Let us call that value
"support." [H|E] has that value just when the particular account of
hypothetico-deductive support at issue says that E supports H.
Confirmational support, however, more naturally comes in degrees. We can certainly imagine augmentations
of the above rules that give a quantitative measure of support. For
example, in hypothetical deduction, we might reward an hypothesis H
if it entails the evidence E in fewer steps. That means that H is
closer to E. There might also be some reward for parsimony.

What do we mean by "number of
steps"? In a Boolean algebra formed by the logical closure of
mutually exclusive atoms a, b, c, ..., we would define one step as
the addition of one atom. So the inference from

a to (a or b)

is a one-step inference. The inference from

(a or b) to (a or b or c or d or e or f)

is a four-step inference.

This suggests the following rule

Evidence E is confirmed
to degree H with strength [H|E] = 1/number

if

H entails E in a deductive inference of
number steps.

This rule is still rather primitive. However no particular
rule is at issue here. My concern in displaying this parade of examples is
merely to make plausible that deductively definable logics of induction are
not exceptional. They are, in effect, the sort of
thing that is discussed routinely in the inductive inference literature.